Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems

Abstract

The Inexact Restoration method for Euler discretization of state and control constrained optimal control problems is studied. Convergence of the discretized (finite-dimensional optimization) problem to an approximate solution using the Inexact Restoration method and convergence of the approximate solution to a continuous-time solution of the original problem are established. It is proved that a sufficient condition for convergence of the Inexact Restoration method is guaranteed to hold for the constrained optimal control problem. Numerical experiments employing the modelling language AMPL and optimization software Ipopt are carried out to illustrate the robustness of the Inexact Restoration method by means of two computationally challenging optimal control problems, one involving a container crane and the other a free-flying robot. The experiments interestingly demonstrate that one might be better-off using Ipopt as part of the Inexact Restoration method (in its subproblems) rather than using Ipopt directly on its own.

Appendix

Proof of Proposition 2.1

It is not difficult to show that ∥V _α (t)V _α (t)^T ζ∥≥β∥ζ∥ if, and only if, V _α (t)V _α (t)^T is nonsingular. This, in turn, is equivalent to the condition that V _α (t) has full row rank. First, define the submatrices

$$V_{\alpha}^1(t) := \left[\begin{array}{c@{\quad}c@{\quad}c} \varTheta(t) &\ U_{\alpha}(t) &0 \end{array}\right] \quad\mbox{and}\quad V_{\alpha}^2(t) := \left[\begin{array}{c@{\quad}c@{\quad}c} \varUpsilon(t)B(t)&0&T_{\alpha}(t) \end{array}\right] . $$

In what follows, we will not show, whenever appropriate, dependence of variables on t, for simplicity in appearance. Furthermore, all arguments below will be made for some fixed t.

Denote the (2i−1)st row of $V_{\alpha}^{1}(t)$ by r _2i−1, i=1,…,m. Then, clearly,

$$r_{2i-1} = e_i + \min\bigl\{0,u_i^* - \overline {b}_i + \alpha\bigr\} e_{m+2i-1} $$

and

$$r_{2i} = -e_i + \min\bigl\{0,-u_i^* + \underline {b}_i + \alpha\bigr\} e_{m+2i} , $$

where e _i is the ith standard basis vector in ℝ^3m+2n. Let p _2j−1, j=1,…,n, denote the (2j−1)st row of $V_{\alpha}^{2}(t)$. Then

$$p_{2j-1} = \sum_{k=1}^m \frac{\partial f_j}{\partial u_k}\bigl(x^*,u^*\bigr) e_k + \min\bigl \{0,x_j^* - \overline {a}_j + \alpha\bigr\} e_{3m+2j-1} $$

and

$$p_{2j} = -\sum_{k=1}^m \frac{\partial f_j}{\partial u_k}\bigl(x^*,u^*\bigr) e_k + \min\bigl \{0,-x_j^* + \underline {a}_j + \alpha\bigr\} e_{3m+2j} . $$

Suppose that $i\in \widetilde {J}_{u}:=J_{u}\backslash \widehat {J}_{u}$. Then $\underline {b}_{i} < u_{i}^{*} < \overline {b}_{i}$ and $\min\{u_{i}^{*}-\underline {b}_{i},\overline {b}_{i}-u_{i}^{*}\}>0$. Next, choose α such that $0<\alpha < \widetilde {\alpha }_{i} := \min\{u_{i}^{*}-\underline {b}_{i},\overline {b}_{i}-u_{i}^{*}\}$. Since $\min\{0,u_{i}^{*} - \overline {b}_{i}+\alpha\}\neq0$, $\min\{0,-u_{i}^{*} + \underline {b}_{i}+\alpha\}\neq0$, and e _m+2i−1≠e _m+2i , the rows r _2i−1 and r _2i are linearly independent.

Suppose that $j\in \widetilde {J}_{x}:=J_{x}\backslash \widehat {J}_{x}$. Via similar arguments, α can be chosen such that $0<\alpha < \widetilde {\beta }_{j} := \min\{x_{j}^{*}-\underline {a}_{j},\overline {a}_{j}-x_{j}^{*}\}$, resulting in $\min\{0,x_{i}^{*} - \overline {a}_{i}+\alpha\}\neq0$, $\min\{0,-x_{i}^{*} + \underline {a}_{i}+\alpha\}\neq0$, and thus linearly independent p _2j−1 and p _2j .

Suppose that $i\in \widehat {J}_{u}$. Then either $u_{i}^{*}=\overline {b}_{i}$ or $u_{i}^{*}=\underline {b}_{i}$ (only one side of the constraint is active at a given t). So $\max\{u_{i}^{*}-\underline {b}_{i},\overline {b}_{i}-u_{i}^{*}\} > 0$. Choose α such that $0<\alpha < \widehat {\alpha }_{i} := \max\{u_{i}^{*}-\underline {b}_{i},\overline {b}_{i}-u_{i}^{*}\}$. Then one of $c_{i}^{1}:=\min\{0,u_{i}^{*} - \overline {b}_{i}+\alpha\}$ and $c_{i}^{2}:=\min\{0,-u_{i}^{*} + \underline {b}_{i}+\alpha\}$ is zero and the other nonzero. Let the index set of the rows with $c_{i}^{1}=0$ or $c_{i}^{2}=0$ be denoted by $\widehat {J}_{u}^{0} = \{k_{1},\ldots,k_{q}\}$ and the index set of the rows with c ₁≠0 or c ₂≠0 by $\widehat {J}_{u}^{1} = \{\ell_{1},\ldots,\ell_{q}\}$. Then, for $i\in \widehat {J}_{u}$ and $\ell\in \widehat {J}_{u}^{1}$, we get

$$r_\ell = \pm e_i + \widetilde {\gamma }_i^u e_{m+\ell} , $$

where $\widetilde {\gamma }_{i}^{x}\neq0$, and the sign of the first term depends on which side of a constraint is active. For $i\in \widehat {J}_{u}$ and $k\in \widehat {J}_{u}^{0}$, we have

$$r_k = \pm e_i . $$

Suppose that $i\in \widehat {J}_{x}$. Proceeding similarly, we choose α such that $0<\alpha < \widehat {\beta }_{i} := \max\{x_{i}^{*}-\underline {a}_{i},\overline {a}_{i}-x_{i}^{*}\}$. Let the index set of the rows corresponding to the active part of the two-sided constraints be denoted by $\widehat {J}_{x}^{0} = \{\widehat {k}_{1},\ldots,\widehat {k}_{s}\}$ and the index set of the rows corresponding to the inactive part of the two-sided constraints be denoted by $\widehat {J}_{x}^{1} = \{\widehat {\ell }_{1},\ldots,\widehat {\ell }_{s}\}$. Then, for $j\in \widehat {J}_{x}$ and $\widehat {\ell }\in \widehat {J}_{x}^{1}$, we get

$$p_{\widehat {\ell }} = \pm \sum_{\widetilde {k}=1}^m \frac{\partial f_j}{\partial u_{\widetilde {k}}}\bigl(x^*,u^*\bigr) e_{\widetilde {k}} + \widetilde {\gamma }_i^x e_{3m+\widehat {\ell }} , $$

where $\widetilde {\gamma }_{i}^{x}\neq0$, and, for $j\in \widehat {J}_{x}$ and $\widehat {k}\in \widehat {J}_{x}^{0}$, we have

$$p_{\widehat {k}} = \pm \sum_{\widetilde {k}=1}^m \frac{\partial f_j}{\partial u_{\widetilde {k}}}\bigl(x^*,u^*\bigr) e_{\widetilde {k}} . $$

The rows r _2i−1 and r _2i for all $i\in \widetilde {J}_{u}$, p _2j−1 and p _2j for all $j\in \widetilde {J}_{x}$, r _k for all $k\in \widehat {J}_{u}^{1}$, and $p_{\widehat {k}}$ for all $\widehat {k}\in \widehat {J}_{x}^{1}$, are linearly independent, because each of these rows contains a nonzero element, which is either in the diagonal of the submatrix U _α or in the diagonal the submatrix T _α .

Now let us look at the remaining rows: r _ℓ , for all $\ell\in \widehat {J}_{u}^{0}$, have a nonzero element only in the submatrix Θ, and $p_{\widehat {\ell }}$, for all $\widehat {\ell }\in \widehat {J}_{x}^{0}$, have nonzero elements only in the submatrix ϒB. Therefore, V _α has full row rank if, and only if, the block

$$\left [\begin{array}{c} r_{\ell_1} \\ \vdots \\ r_{\ell_q} \\ \cdots\cdots\cdots\cdots \\ p_{\widehat {\ell }_1} \\[2mm] \vdots \\[1mm] p_{\widehat {\ell }_s} \end{array} \right ] $$

has linearly independent rows, which is a condition equivalent to that in (10). □